Prompt programming treats LLM prompts as software components with typed interfaces. Through a literature survey of 15 recent works (2023-2025), we observe a consistent trend: type systems are central to emerging prompt programming frameworks. However, there are gaps in constraint expressiveness and algorithms. To address it, we introduce the notion of $\lambda$ Prompt, a dependently typed calculus with probabilistic refinements for syntactic/semantic constraints. While not yet a full calculus, our formulation motivates a type-theoretic foundation for prompt programming. Our catalog of 13 constraints reveals underexplored areas in constraint expressiveness (C9–C13). To address algorithmic gap, we propose a constraint-preserving optimization rule. Finally, we outline research directions on prompt program compiler.